The agency estimates that it collects $458 billion a year less in taxes from all Americans than the government is actually due. Most of that “tax gap” is income that goes unreported, and the biggest chunk of it, by far — $125 billion — is individual business income.

Taxpayers in this category, primarily sole proprietors, pay taxes on the money their operations make through their personal returns. Thus, their cash flows can be particularly opaque.

Take owners like Rebeca Mojica, a chainmaille jewelry designer in West Hollywood, Calif., who was audited in 2011. Her sales fluctuate significantly from year to year, and she takes some payments in cash. Were she so inclined, she could easily hide a chunk of that income.

The dreaded audit is the main way the I.R.S. catches scofflaws and ferrets out unreported income, but it is a time-consuming and imperfect tool. Short on resources, the agency collected just $7.3 billion from audits last year, its lowest total in 13 years.

What the I.R.S. really wants is for business owners to voluntarily pay more of what they owe. But 63 percent of “low visibility” income, the kind that isn’t captured by outside parties on tax information documents, is not disclosed on tax forms, the agency says.

So for the last four years, the Taxpayer Advocate Service, an independent office within the I.R.S., has been running studies to help it figure out how more small-business owners who pay their taxes can be persuaded to report their earnings more accurately.

One finding suggests that audits, the agency’s most powerful compliance tool, seem to have little lasting deterrent effect on tax cheats, and could even backfire for honest taxpayers.

That discovery, and others, could help inform the agency’s future collection techniques.

For one, those suspected of tax dodging tend to cluster in certain geographic areas. The agency mapped out 365 communities with notably low tax-compliance rates among sole proprietors. Many were based in the South and West. California alone makes up a full third of the list; Georgia and Texas were also heavily represented.

(The researchers tried to compile a similar data-set of high-compliance places where business owners are likely to report their earnings accurately, but it found only three. Those in Mott Haven, a neighborhood in the South Bronx; West Somerville, Mass.; and Portersville, Ind., appear to be unusually law-abiding.)

What is behind the clusters? Community norms play a major role, the tax researchers theorized. Business owners who were less compliant in paying their taxes were more likely to be active in civic groups and religious congregations, and they “appear to exhibit a stronger association with local institutions than national ones such as the federal government.”

In other words, if people at your neighborhood potlucks or trade group meetings take a dim view of Washington, you might be more inclined to do some fudging at tax time.

Another surprise was the effectiveness of audits.

Self-employed individuals who went through an audit and were found to be clean reported less income in subsequent years, a different study found. The drop wasn’t small: Three years after their audits, the study’s test group of taxpayers reported 35 percent less in taxable income than a control group of similar taxpayers who had not been audited.

The researchers could only guess at a cause. The audit may have been a discouraging experience and sapped the subject’s “tax morale.” Or perhaps it inadvertently offered insight into previously unknown tactics for both legal tax avoidance and illegal tax evasion.

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Most audits are not random. The I.R.S. has a secret algorithm that it uses to calculate how likely each taxpayer is to have unreported income. Those with high scores are more likely to be audited — and once the auditors start digging, they usually find things. Of the 1.2 million individual returns that the agency audited (including sole proprietorships) in 2014, only 13 percent emerged without any tax adjustments.

For those who dodge their taxes and get caught, the sting seems to fade fast. In the years right after an audit, taxpayers who had to make additional payments appeared to become a bit more compliant, but the effect diminished over time and disappeared entirely by Year 5, another study found.

“Any initial impact of the audit on compliance is short-lived,” the researchers concluded.

That doesn’t surprise Fred Daily, a Florida tax lawyer who specializes in audits and tax crimes.

“I’ve had people who got caught by the I.R.S. and got serious damage — hundreds of thousands of dollars of damage — and at first, they’re like people just out of jail: ‘I’m never going back again!’ A few years later, they’re right back to doing what they had done before,” he said. “People’s character doesn’t change.”

Still, Mr. Daily says that only a minority of those he encounters actively intend to cheat. Many people, especially business owners, run into problems in an audit simply because they kept bad records or did not understand all of their tax obligations.

“It’s so easy to call yourself a sole proprietor; just put up a website or print a business card,” he said. “They don’t realize all the requirements, like quarterly estimated reporting and the self-employment tax. The tax laws are impenetrable.”

But even for law-abiding owners, an audit is unnerving and time-consuming. Ms. Mojica, the jewelry designer, estimates that she spent $500 in accounting fees and 15 hours of her and her staff’s time dealing with her 2011 audit. The process lasted seven months.

In the end, she owed nothing — which provided an unanticipated psychological boost.

“I had this feeling of, ‘Yay! I’m doing all right!’” Ms. Mojica said. “I also felt motivated to keep even better records. I spent an hour or two trying to find information on a very small number of transactions, and I could have been spared that if we’d kept a better paper trail.”

The good news for business owners is that audits are extremely rare. Around 1.5 percent of self-employed taxpayers are audited each year, the agency says.

In any case, businesses should always brace for the possibility of an audit. “Document everything” is the advice Vanessa Kruze, an accountant who focuses on start-ups, gives her clients. Expense reports draw particular scrutiny from auditors, and using a program like Expensify to track receipts makes it much simpler to back up claims, she said.

Mr. Daily, who has been steering clients through audits for 35 years, said the biggest surprise to him is how rarely the I.R.S. checks back on the worst offenders.

The agency’s budget cuts have taken a noticeable toll, he said.

“It’s really counterintuitive how they do not stay on some of these people and audit them year after year,” Mr. Daily said. “I’ve had people where, as we finish an audit, I’ve said, ‘This very likely isn’t the last time you will see the I.R.S.’ — and actually, it is.”

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